Optimal area triangulation

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Abstract

Given a set of points in the Euclidean plane, we are interested in its triangulations, i.e., the maximal sets of non-overlapping triangles with vertices in the given points whose union is the convex hull of the point set. With respect to the area of the triangles in a triangulation, several optimality criteria can be considered. We study two of them. The MaxMin area triangulation is the triangulation of the point set that maximizes the area of the smallest triangle in the triangulation. Similarly, the MinMax area triangulation is the triangulation that minimizes the area of the largest area triangle in the triangulation. In the case when the point set is in a convex position, we present algorithms that construct MaxMin and MinMax area triangulations of a convex polygon in $O(n^2log{n})$ time and $O(n^2)$ space. These algorithms are based on dynamic programming. They use a number of geometric properties that are established within this work, and a variety of data structures specific to the problems. Further, we study polynomial time computable approximations to the optimal area triangulations of general point sets. We present geometric properties, based on angular constraints and perfect matchings, and use them to evaluate the approximation factor and to achieve triangulations with good practical quality compared to the optimal ones. These results open new direction in the research on optimal triangulations and set the stage for further investigations on optimization of area.